The Crows led by 19 points at three-quarter time but slipped behind in the final term, before prevailing 14.6 (90) to 9.14 (68) on Saturday night.
Adelaide will travel to face reigning premiers Brisbane in a semi-final on Saturday night, with Fremantle to host Geelong in Perth on Friday night in the other semi.
The win eases pressure on Crows coach Matthew Nicks - after a straight-set finals exit last year, his tenure reportedly hinged on beating the Bulldogs.
"I've been around long enough to understand the industry and the narrative and why why things come up about coaches," Nicks said.
"I understand how that works and we've worked in that space for a long time, so it was business as usual."
Nicks would have been relatively comfortable at three-quarter time, but eight minutes later his side was down by a point after a Bulldogs fightback of three rapid-fire goals.
Key Crows forwards Riley Thilthorpe, Taylor Walker, in his last game at Adelaide Oval before retiring, and Darcy Fogarty responded with clutch goals to deliver the South Australian club a first playoff win since the 2017 preliminary final.
Star Crow Izak Rankine, who missed last year's finals when serving a four-game ban for a homophobic slur, was instrumental with 34 disposals, seven clearances and two goals.
Defenders Wayne Milera (27 disposals) and Josh Worrell (34) were outstanding, Josh Rachele kicked three goals, and Rankine, Fogarty, Ben Keays and Brayden Cook booted two each.
The Dogs' prime movers Ed Richards (27 touches, two goals), Marcus Bontempelli (27) and Joel Freijah (24) were busy, as was Bailey Williams, while Arthur Jones and Will Lewis also kicked two goals apiece.
"We always feel like we fall short when we don't win the whole thing," Bulldogs coach Luke Beveridge said.
"(But) we have made some progress, we really have ... our players surprised us at times."
The Bulldogs suffered a late withdrawal in bizarre circumstance when defender Michael Sellwood was hit in the head by a football during the on-field warm-up and concussed.
Adelaide led 3.1 to 1.7 at quarter-time and stretched to a 12-point break early in the second term before the Bulldogs rallied with three goals in four minutes.
The spurt included a Jones strike after a dubious free kick that enraged the 51,370-strong pro-Crows crowd - their mood didn't improve when the Dog blew them kisses after converting.
The visitors were six points - their biggest lead of the game - but  Adelaide responded with the next three goals for a 7.2 to 5.8 edge at halftime.
The Crows seemingly made a decisive move with two goals to none in the third term to create a 19-point buffer at the last change.
But the Dogs hit the front with three goals and two behinds, including an Aaron Naughton shot that hit the post.
The Naughton poster was followed by Adelaide's game-breaking retort, with big guns Thilthorpe, Walker and Fogarty firing the Crows into the semi-finals.
Rankine, with a remarkable snap from a near-impossible angle, and Rachele then iced victory with late goals.
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